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& IWA Publishing 2011 Hydrology Research 9 42.2–3 9 2011
150
Real-time flood stage forecasting by Variable Parameter Muskingum Stage hydrograph routing method Muthiah Perumal, Tommaso Moramarco, Silvia Barbetta, Florisa Melone and Bhabagrahi Sahoo
ABSTRACT The application of a Variable Parameter Muskingum Stage (VPMS) hydrograph routing method for real-time flood forecasting at a river gauging site is demonstrated in this study. The forecast error is estimated using a two-parameter linear autoregressive model with its parameters updated at every routing time interval of 30 minutes at which the stage observations are made. This hydrometric data-based forecast model is applied for forecasting floods at the downstream end of
Muthiah Perumal (corresponding author) Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee – 247 667, India Phone: þ 91-1332-285817 (Work); þ 91-1332-285011 (Home), Fax: þ 91-1332-285236, 273560 (Work) E-mail:
[email protected]
a 15 km reach of the Tiber River in Central Italy. The study reveals that the proposed approach is able to provide reliable forecast of flood estimate for different lead times subject to a maximum lead time nearly equal to the travel time of the flood wave within the selected routing reach. Moreover, a comparative study of the VPMS method for real-time forecasting and the simple
Tommaso Moramarco Silvia Barbetta Florisa Melone Research Institute for Geo-Hydrological Protection, National Research Council, 06128 Perugia, Italy
stage forecasting model (STAFOM), currently in operation as the Flood Forecasting and Warning System in the Upper-Middle Tiber River basin of Italy, demonstrates the capability of the VPMS model for its field use. Key words 9 compound channel, flood, hydrograph, Muskingum stage, real-time forecasting
variable parameter
Bhabagrahi Sahoo Soil and Water Conservation Engineering, ICAR Research Complex for NEH Region, Nagaland Centre, Jharnapani, Medziphema – 797 106, Nagaland, India Formerly at Department of Hydrology, Indian Institute of Technology Roorkee 247 667, India
INTRODUCTION Many communities owe much of their prosperity to advan-
is an important non-structural measure for flood damage
tages offered by adjacent and nearby streams, the more
reduction and for minimising flood-related deaths and,
important being adequate commercial and municipal water
hence, its implementation as an effective tool requires accu-
supplies, navigation, power development and recreation.
rate flood forecasting with sufficient lead time. Hence, it is
Adverse effects, however, are experienced when high flows
essential that flood forecasting methods should be physically
occur in the form of floods causing loss of life and damage to
based, less data intensive and, over and above, should be
property which have to be mitigated by employing economic-
easily understood by the field engineers.
ally feasible structural measures such as levees, flood walls
Every flood forecasting model operates in two modes: the
and channel improvement. However, these types of measures
simulation mode, and the operation mode (on-line forecas
cannot eliminate completely the hydraulic risk, given the
ting). A flood forecasting model in the simulation mode
impossibility of building larger and larger structures to cope
attempts to reproduce the response of the system for past
with extremely low probability events. Therefore, an impor-
recorded precipitation or upstream input flow. The response
tant role remains for non-structural measures to be com-
of the model is compared with the recorded response at the
pared, evaluated and actuated in real time. Flood forecasting
section of forecasting interest and, if they do not match, either
doi: 10.2166/nh.2011.063
151
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
the model structure is changed or the parameters are modified until the match is satisfactory. Once the structure of the
Hydrology Research 9 42.2–3 9 2011
VARIABLE PARAMETER MUSKINGUM STAGE– HYDROGRAPH ROUTING METHOD
model and its parameters have been identified during the calibration phase, the model can be used for forecasting
The physically based VPMS hydrograph routing method was
purposes and it is said to be used in operational mode.
developed by Perumal & Ranga Raju (1998a, b) directly from
While the basic structure of the model is not changed in the
the Saint Venant equations. The form of the routing equation
operational mode, the parameters need to be changed to
developed is the same as that of the Muskingum method,
reflect the current catchment conditions due to the variation
replacing the discharge variable by the stage variable, which
of the input.
is the reason for adherence to the term ‘‘Muskingum’’.
Typically, the flood forecasting models have two compo-
Further, the parameters vary at every routing time interval
nents: the deterministic flow component and the stochastic
and they are related to the channel and flow characteristics
flow component. While the former is determined by the
by the same relationships as established for the physically
hydrologic/hydraulic model, the latter is determined based
based Muskingum method (Apollov et al. 1964; Cunge 1969;
on the residual (error) series of the difference between the
Dooge et al. 1982; Perumal 1994a, b). The detailed develop-
forecasted flow for a specified lead time and the correspon-
ment of the method can be found in Perumal & Ranga Raju
ding observed one. The residual series reflects both the model
(1998a, b) and Perumal et al. (2007). Only the equations
error, due to the inability of the model used for forecasting to
relevant to this study are presented here.
correctly reproduce the flow process, and the observational
Using the Approximate Convection–Diffusion equation
error while measuring the flow. It is imperative, therefore, to
of the following flow depth formulation (Perumal & Ranga
use an appropriate approach to reduce the model error. The
Raju 1999):
adaptive parameter estimation methods employing the Kalman filtering technique may not be worth the effort for real-time flood forecasting (Ahsan & O’Connor 1994; Huang 1999), when the hydrological model employed for forecasting is grossly inadequate to simulate past recorded floods. In such a scenario, the application of the simplified physically based model like the variable parameter Muskingum stage (VPMS) hydrograph routing method along with a simple error updating technique may be found useful for real-time flood forecasting at a river gauging site.
@y @y þc ¼0 @t @x
ð1Þ
the Muskingum-type routing equation can be arrived at as (Perumal 1998a) yu y d ¼ K
d ½yd þ yðyu yd Þ dt
ð2Þ
where yu and yd denote the flow depths at the upstream and downstream sections of the Muskingum reach, respectively. The travel time K can be expressed as
The analysis presented here focuses on this specific aspect by studying the use of a VPMS routing method as a compo-
K¼
nent model of a hydrometric data-based deterministic fore-
Dx c3
ð3Þ
casting model. It will be shown that the use of a physically
where Dx is the length of the Muskingum reach and c3 is the
based component model in a forecasting model enables the
wave celerity.
use of a simple stochastic error updating model to estimate the forecast additive error. The proposed forecasting model is
The weighting parameter y, after neglecting the inertial terms, can be expressed as
tested by considering several flood events that occurred along a 15 km river reach of the Tiber River, in Central Italy, bounded by Pierantonio and Ponte Felcino gauging stations
y¼
1 Q3 2 2S0 ð@A/@yÞ3 c3 Dx
ð4Þ
and comparing its accuracy with that of a simple Stage Forecasting Model (STAFOM) currently in operation as the
The subscript 3 attached to different variables in Equa-
Flood Forecasting and Warning System in the Upper-Middle
tions (3) and (4) denotes the evaluation of these variables at
Tiber River basin.
section 3, at which the normal discharge corresponding to the
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M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
Hydrology Research 9 42.2–3 9 2011
flow depth at the middle of the Muskingum reach passes
channel and a floodplain channel as shown in Figure 2. It has
during unsteady flow (see Figure 1); Q denotes the discharge;
been shown therein that the wave celerity corresponding to
S0 is the bed slope and @A/@y is the top width of the water
flow in the main channel, cmain, is expressed as
surface. Using Equations (3) and (4) in Equation (2) and expres-
cmain
sing it as a difference equation leads to a form similar to that
5 2 Rmain ð@Pmain /@yÞ Qmain ¼ 3 3 ð@Amain /@yÞ Amain
ðyoym Þ
ð7Þ
of the Muskingum routing equation, but using flow depth as
where ym is the main channel depth; Amain, Pmain, and Rmain
the operating variable, and it is expressed as
represent the flow area, the wetted perimeter and the hydrau-
yd;jDt ¼ C1 yu;jDt þ C2 yu;ðj1ÞDt þ C3 yd;ðj1ÞDt
ð5Þ
where yu,jDt and yd,jDt denote the observed upstream and the estimated downstream flow depths at time jDt, respectively;
lic radius for the main channel, respectively; and Qmain is the discharge of the main channel section. The wave celerity for flow in the compound channel is computed as (Perumal et al. 2007)
and yu,(j1)Dt and yd,(j1)Dt denote the observed upstream and downstream flow depths at time (j–1)Dt, respectively. The notation Dt is the routing time interval, and the coefficients
ccompound ¼
C1, C2 and C3 are expressed as þ Ky þ 0:5Dt C1 ¼ Kð1 yÞ þ 0:5Dt
ð6aÞ
Ky þ 0:5Dt Kð1 yÞ þ 0:5Dt
ð6bÞ
C2 ¼
C3 ¼
5 @Amain vmain 3 @y
=
5 @A1 2 A1 @P1 v1 3 @y 3 P1 @y
5 @A2 2 A2 @P2 v2 þ 3 @y 3 P2 @y
@Acompound @y
= @A @y
compound
= @A @y
compound
ð8Þ
ðy4ym Þ
where vmain denotes the velocity of flow in the main channel; v1 and v2 are the flow velocities in the floodplains 1 and 2
Kð1 yÞ 0:5Dt Kð1 yÞ þ 0:5Dt
ð6cÞ
(shown in Figure 2), respectively; A1, P1, A2 and P2 denote the flow area and wetted perimeter of the two floodplains, respectively; and Acompound is the total flow area of the
It has been shown by Perumal et al. (2007) that the VPMS method can be applied for routing in a uniform compound trapezoidal cross-section channel reach consisting of a main
compound channel. The flow velocities in the main channel and in floodplains 1 and 2 of the compound channel are evaluated as
vmain
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi S0 1 @y ð2/3Þ ðRmain Þ ¼ 1 S0 @x n
ð9aÞ
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi S0 1 @y ð2/3Þ v1 ¼ ðR1 Þ 1 n S0 @x
M
ð9bÞ
3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi S0 1 @y ð2/3Þ ðR2 Þ 1 v2 ¼ S0 @x n
2
yu Qu yM
QM
ð9cÞ
Q3 y3
yd Qd
Δx/2
@y yd yu ¼ Dx @x
ð9dÞ
where Rmain, R1 and R2 denote the hydraulic radius of the Δx
L
main channel section and of the floodplains 1 and 2 of the compound channel section, respectively; and n is Manning’s
Figure 1 9 Definition sketch of the stage–hydrograph routing method.
roughness coefficient.
153
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
Hydrology Research 9 42.2–3 9 2011
bm+2ymz1
(a)
2 1
1
bf z2 MAIN
1
z1 bm
Elevation (m)
(b) 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 –0.5 0 –1.0
Pierantonio section Ponte Felcino section trapezoidal section
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Distance (m)
Figure 2 9 a) Prismatic compound channel section used for the actual river conceptualization; it is made up of a main channel section (shaded) and two floodplain channels (sections 1 and 2). b) Cross-sections of the Upper Tiber River at Pierantonio (upstream) and Ponte Felcino (downstream) gauging stations with the optimized trapezoidal channel section.
VPMS MODEL FOR REAL-TIME APPLICATION
yˆ d;ðj1ÞDtþTL . However, only the last one is known, being the forecast estimate of the downstream stage assessed at the
In order to apply the VPMS method for real-time fore-
previous time of forecast, (j–1)Dt. Therefore, in order to apply
casting purpose, the routing equation given by Equation (5)
Equation (10) for estimation of yˆ d;ðjDtþTL Þ , the following
has to be suitably modified considering a forecast lead time,
assumption has to be made based on no-model hypothesis:
TL, as
yˆ u;jDtþTL ¼ yˆ u;ðj1ÞDtþTL ¼ yu;jDt
yˆ d;ðjDtþTL Þ ¼ C1 yˆ u;ðjDtþTL Þ þ C2 yˆ u;ðj1ÞDtþTL þ C3 yˆ d;ðj1ÞDtþTL þ ef;ðjDtþTL Þ
ð11Þ
where yu,jDt is the last upstream observed stage. ð10Þ
Using Equation (11)) in Equation (10), the final forecasting model is expressed as
where yˆ denotes the forecast stages, and ef;ðjDtþTL Þ is the error of forecast, that is, the difference between the observed stage and the corresponding forecasted stage at the site of forecast interest. It can be inferred from Equation (10) that at the time
yˆ d;ðjDtþTL Þ ¼ C1 yu;jDt þ C2 yu;jDt þ C3 yˆ d;ðj1ÞDtþTL þ ef;ðjDtþTL Þ
ð12Þ
of forecast jDt, in order to get the forecast estimate of the downstream stage with a lead time TL, three different forecast
In Equation (12), the minimum TL is Dt, the routing time
quantities should be available, i.e., yˆ u;ðjDtþTL Þ , yˆ u;ðj1ÞDtþTL and
interval at which the stage measurements are made, and this
154
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
Hydrology Research 9 42.2–3 9 2011
corresponds to one time interval ahead forecast. The maximum
was made to study the sensitivity of the order of the stochastic
lead time interval that can be adopted depends on the accuracy
error model and the initial warm-up period on the estimates
of the obtained forecast and that may nearly correspond to the
of the forecast. The parameters a1 and a2 are updated in real
travel time of the upstream discharge to arrive at the site of
time on the basis of the last available stage observations.
forecast interest. The use of a larger TL beyond this approximate travel time would lead to poorer accuracy of the forecast. In order to estimate ef;ðjDtþTL Þ in Equation (12), an error
FIELD APPLICATION
updating model also needs to be developed for estimating the forecast error, which when added to the model estimated
The proposed forecasting model consisting of the VPMS
forecast for a given lead time would yield the final forecasted
routing method, as the basic model, and the second-order
stage at the site of interest. Note that different error updating
linear autoregressive model, as the error updating model, is
techniques of varied complexities such as Kalman filtering
applied for forecasting the flow stage in a 15 km reach along
(Gelb 1974; Ahsan & O’Connor 1994; Neal et al. 2007), the
the Tiber River, in Central Italy. The selected reach is
auto-regressive moving average (ARMA) model (Box &
bounded by Pierantonio and Ponte Felcino gauging stations
Jenkins 1970), and Artificial Neural Networks (e.g., Babovic
and has an average bed slope S0 of 0.0016 and a Manning
et al. 2001) are available in the literature. Refsgaard (1997) has
roughness coefficient n ¼ 0.039.
provided the classification and review of different error
Note that the approximation of the VPMS method for
updating procedures currently used in real-time flood fore-
routing a given stage hydrograph in a river reach requires the
casting. However, for simplicity, it is proposed to use a
use of an equivalent prismatic channel reach; this involves the
second-order linear autoregressive error updating model of
approximation of the actual river reach sections at the two
the following form for forecasting the error at time (jDt þ TL):
ends to an equivalent prismatic section with a one-to-one relationship established between the flow depth of the actual
ef;ðjDtþTL Þ ¼ a1 eobs;jDt þ a2 eobs;ðj1ÞDt þ EðjDtþTL Þ
ð13Þ
section of a given flow area with the corresponding flow depth of the prismatic channel section of the same flow area. Based
where eobs,jDt and eobs,(j1)Dt are the forecasting errors esti-
on the surveyed cross-sections at the ends of the actual river
mated at time jDt and (j–1)Dt, respectively, and EðjDtþTL Þ is the
reach, it was considered appropriate to approximate the actual
random error (white noise).
reach by a compound trapezoidal section reach. Accordingly,
Forecasting using Equation (13) can be made only after
the surveyed cross-sections of the actual reach were over-
the lapse of certain initial period of the forecasting event,
lapped and a two-stage trapezoidal compound section geo-
known as the warm-up period. The difference between the
metry was assessed paying particular attention to the flow area
observed stage and the VPMS routed stage in the warm-up
reproduction. In particular, once the floodplain level, ym, (see
period is considered as the actual error and its series is
Figure 2) is assumed on the basis of the properties of the two
assumed to be stochastic in nature. The initial parameters
channel ends, the section parameters bm, bf , z1 and z2 (see
a1 and a2 of the error update model are assessed using this
Figure 2 for symbols) are assessed by minimizing the mean
error series estimated in the warm-up period. The duration of
square error in the real mean flow area estimate (see Perumal
initial warm-up period considered for developing the error
et al. 2010). Based on this criterion, a compound trapezoidal
update model should not be too long to avoid that the
section with bm ¼ 27.31 m, ym ¼ 5.0 m, bf ¼ 57.6 m, z1 ¼ 1.98
forecasting exercise becomes of no practical use for forecast-
and z2 ¼ 3.8 was identified. Therefore, the relationships
ing the given event, and, at the same time, it should not be too
between the actual flow depths and the equivalent trapezoidal
short resulting in numerical problem while estimating the
section ones at the channel ends were developed in order to
parameters a1 and a2 using the least squares approach.
have the same value of the actual mean flow area, yielding
However, in this study, the error updating model given by Equation (13) has been applied without generating the random error component. It may be noted that no attempt
yutrap ¼ 0:916 yuactual þ 0:065
ð14Þ
ydtrap ¼ 1:079 ydactual 0:067
ð15Þ
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
155
Hydrology Research 9 42.2–3 9 2011
where yu-trap and yd-trap are the equivalent upstream and
criteria: (1) the Nash–Sutcliffe (NS) efficiency coefficient
downstream flow depths in the trapezoidal channel section
(Nash & Sutcliffe 1970) and (2) the Persistence Criterion
corresponding to the flow depths yu-actual and yd-actual in the
(PC). As the NS coefficient is well known in hydrological
actual river section. Using the upstream section relation-
literature (ASCE 1993), only the Persistence Criterion is
ship, the observed stage hydrograph of any event was
explained here. It compares the prediction of the proposed
converted to equivalent trapezoidal section stage hydro-
model against that obtained by the no-model, which assumes a
graph to enable the routing using the VPMS method and,
steady state over the forecasting lead time, and is evaluated as
using the relationship (yd-actual ¼ 0.927 yd-trap þ 0.062) developed on the basis of the downstream site properties, the routed hydrograph of the equivalent trapezoidal section
P PC ¼
1P
i ðyiDt
i
yˆ iDt Þ2
ðyiDt yðiDtTL Þ Þ2
! 100
ð16Þ
was converted to the actual end section estimated hydrowhere y and yˆ denote the observed and the forecasted flow
graph. To study the applicability of the proposed forecasting
depth values, respectively.
model, 12 flood events recorded concurrently at Pierantonio
Further, to investigate the reliability of the proposed
and Ponte Felcino stations were used. The details of these
VPMS model for flood forecasting a comparative study
events, each recorded at half -hour intervals, are shown in
between the VPMS solution and the corresponding stage
Table 1, where also the details of wave travel time, percentage
hydrographs forecasted by STAFOM (Moramarco et al.
of lateral flow and actual and equivalent trapezoidal peak
2006; Barbetta et al. 2008), the model currently in operation
flow depths at both the stations are reported. As can be seen,
as the Flood Forecasting and Warning System in the Upper-
on the basis of the selected events, the mean flood wave travel
Middle Tiber River basin, was carried out. STAFOM involves
time for the investigated river reach is nearly equal to 1.5
a physically based approach incorporating the lateral flow
hours.
contribution with an additive error component that is
The accuracy of the proposed forecasting model was
updated using the stage observations available in real-time
studied using a warm-up period of 5 hours and considering
(Barbetta et al. 2008). The model requires the estimation of
five different forecast lead times (1.0, 1.5, 2.0, 2.5 and 3.0
four parameters if the downstream rating curve is unknown,
hours). The efficiency of the forecast was evaluated using two
otherwise only two parameters have to be determined.
Table 1 9 Pertinent characteristics of the flood events studied
Pierantonio section
Event
Wave travel time (h)
Lateral inflow (%)
December 96
1.50
1.90
Ponte Felcino section
Actual peak stage (m)
Equivalent trapezoidal peak stage (m)
Actual peak stage (m)
Equivalent trapezoidal peak stage (m)
4.74
4.32
4.22
4.33
April 97
1.50
6.50
5.07
4.62
4.57
4.70
November 97
1.00
5.40
4.22
3.86
3.81
3.90
February 99
2.00
4.40
5.06
4.61
4.52
4.65
December 99
0.00
24.70
2.71
2.52
2.79
2.82
December 00
2.00
Flooding
5.92
5.37
5.25
5.42
April 01
2.00
0.20
3.68
3.38
3.23
3.29
November 05
2.50
Flooding
7.10
6.42
6.92
7.19
3rd December 05
1.00
3.60
5.10
4.64
4.42
4.55
5th December 05
1.00
5.70
5.49
4.99
4.76
4.91
30th
2.00
1.90
4.99
4.54
4.34
4.46
1.50
28.40
2.28
2.14
2.64
2.66
December 05
February 06
156
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
Hydrology Research 9 42.2–3 9 2011
Table 2 9 Forecasting model results provided by the VPMS and STAFOM models for a lead time of 1 hour (err_ypeak ¼ percentage error in peak stage; err_tpeak ¼ error in time to peak stage)
VPMS model Event
December 96 April 97 November 97 February 99
err_ypeak (%)
STAFOM model err_tpeak (h)
NS (%)
PC (%)
err_ypeak (%)
err_tpeak (h)
NS (%)
PC (%)
0.11
1.50
99.80
93.24
0.12
1.50
99.42
78.69
0.11
0.50
99.95
97.97
0.01
0.50
99.77
88.54
1.00
3.00
99.87
96.17
1.46
3.00
99.73
90.52
0.87
0.50
99.90
96.62
0.82
1.00
99.26
72.42
December 99
2.00
1.00
99.78
77.82
2.43
1.00
99.61
58.70
December 00
0.75
1.50
99.80
90.27
0.33
0.00
99.44
69.10
April 01
0.70
0.50
99.63
95.10
1.84
1.00
98.48
77.39
0.06
0.00
99.87
90.46
0.85
0.00
99.60
68.25
3rd December 05
1.40
0.50
99.73
94.93
3.07
1.00
99.45
88.57
5th December 05
0.17
0.50
99.79
93.24
0.37
0.50
99.74
85.20
30th December 05
0.30
0.50
99.91
92.21
0.45
0.50
99.74
76.68
February 06
1.49
1.00
99.62
81.51
0.46
1.00
99.23
58.59
Mean absolute value
0.75
0.92
99.80
91.63
1.02
0.92
99.46
76.05
November 05
RESULTS AND DISCUSSION
results also include the accuracy of peak reproduction, error in time to peak, Nash–Sutcliffe (NS) efficiency and Persis-
Tables 2–6 show the forecasting results provided by both
tence Criterion (PC) efficiency. The two most significant
the proposed approach and STAFOM for the peak flow
events studied herein are characterized by flooding
stage forecast at Ponte Felcino station for all the selected
(December 2000 and November 2005) with flow spilled
flood events and for all the investigated lead times. The
over the main channel, almost in the entire stretch of the
Table 3 9 As Table 2, but for a lead time of 1.5 hours
VPMS model Event
December 96 April 97 November 97 February 99
err_ypeak (%)
STAFOM model err_tpeak (h)
NS (%)
PC (%)
err_ypeak (%)
err_tpeak (h)
NS (%)
PC (%)
0.54
1.00
99.68
95.11
0.86
1.00
99.47
91.59
0.79
0.00
99.87
97.49
0.99
0.00
99.86
96.91
1.89
2.50
99.81
97.32
2.26
2.50
99.73
95.99
0.10
0.50
99.94
99.01
0.02
0.50
99.52
92.23
December 99
2.60
1.50
99.49
75.60
3.22
1.50
99.37
69.29
December 00
0.84
1.00
99.68
92.69
0.35
3.50
99.37
84.59
0.77
0.00
99.60
97.57
1.21
0.00
99.03
93.77
November 05
April 01
0.38
0.00
99.67
89.05
0.80
1.00
99.30
76.01
3rd December 05
0.31
0.50
98.88
90.54
1.09
0.50
99.13
92.13
5th December 05
0.32
0.00
99.57
93.83
0.09
0.00
99.58
93.61
30th December 05
0.91
0.50
99.86
94.71
1.35
0.50
99.75
90.08
February 06
3.50
0.00
98.88
74.48
0.93
1.50
98.26
58.52
Mean absolute value
1.08
0.63
99.58
91.45
1.10
1.04
99.36
86.23
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
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Hydrology Research 9 42.2–3 9 2011
Table 4 9 As Table 2, but for a lead time of 2.0 hours
VPMS model Event
December 96
err_ypeak (%)
STAFOM model err_tpeak (h)
NS (%)
PC (%)
err_ypeak (%)
err_tpeak (h)
NS (%)
PC (%)
0.98
0.50
99.36
94.35
1.23
0.50
99.51
95.63
0.28
2.00
99.42
93.33
0.48
0.50
99.79
97.51
November 97
2.50
3.50
99.44
95.61
3.02
1.50
99.58
96.67
February 99
0.50
0.50
99.64
96.75
0.15
0.50
99.85
98.64
3.48
2.00
99.04
73.18
0.02
2.50
99.26
89.94
April 97
December 99
3.00
2.00
98.8
66.6
December 00
0.20
8.50
99.28
90.26
3.50
0.00
97.85
92.51
0.87
0.50
99.26
97.43
0.65
1.00
99.38
88.29
1.51
0.50
99.15
84.00
April 01 November 05 3rd December 05
1.87
8.00
95.59
78.22
0.65
0.00
97.40
87.01
5th December 05
1.43
3.00
98.6
88.51
0.35
3.00
99.15
92.93
30th
1.27
0.50
99.67
92.78
1.34
0.50
99.76
94.59
February 06
December 05
5.80
0.50
97.25
63.27
9.29
0.50
96.66
55.57
Mean absolute value
1.83
2.50
98.69
86.71
1.87
1.04
99.03
88.59
reach and, also received unaccounted lateral flow (see
and 1.5 hours, whereas for higher TL values, the STAFOM
Table 1). It can be inferred from Tables 2–6 that the
estimates seem to be more reliable.
proposed approach and STAFOM are characterized by
Figures 3–6 show some typical forecasted events for various
similar and high accuracy. However, it can be observed
lead times. The given inflow hydrograph and the corresponding
that the VPMS method provides, on average, more accurate
observed outflow hydrograph are also shown in these figures. It
forecast stage values for a forecasting lead time, TL, of 1.0
is inferred from the results given in Tables 2–6 that, up to a lead
Table 5 9 As Table 2, but for a lead time of 2.5 hours
VPMS model Event
err_ypeak (%)
STAFOM model err_tpeak (h)
NS (%)
PC (%)
err_ypeak (%)
err_tpeak (h)
NS (%)
PC (%)
December 96
3.80
4.50
97.89
87.65
1.77
0.00
98.81
93.34
April 97
0.80
1.50
98.00
84.94
0.47
1.50
99.08
93.50
November 97
5.77
4.00
98.25
91.07
3.30
0.00
98.96
95.00
February 99
3.40
4.00
98.20
89.37
0.93
1.50
99.50
97.17
December 99
3.74
2.50
97.09
46.13
4.51
2.50
98.41
71.26
December 00
3.95
8.50
98.20
83.92
1.25
8.00
98.71
89.13
8.10
1.00
90.91
79.01
4.64
0.50
96.51
92.44
0.94
2.00
98.88
86.38
0.95
1.50
98.47
81.91
April 01 November 05 3rd December 05
8.17
7.50
86.39
55.22
3.21
7.00
92.64
76.77
5th December 05
5.96
4.00
94.82
72.21
3.27
4.00
97.64
87.66
30th December 05
1.63
0.50
98.90
84.15
1.81
2.00
99.47
92.66
February 06
7.98
1.50
94.16
48.03
4.05
1.50
94.48
53.52
Mean absolute value
4.52
3.46
95.97
75.67
2.51
2.50
97.72
85.36
158
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
Hydrology Research 9 42.2–3 9 2011
Table 6 9 As Table 2, but for a lead time of 3.0 hours
VPMS model Event
err_ypeak (%)
December 96
7.95
April 97
2.74
November 97 February 99
STAFOM model err_tpeak (h)
NS (%)
PC (%)
err_ypeak (%)
5.00
94.72
78.00
4.66
7.50
95.53
76.02
1.10
10.10
3.50
96.24
86.48
11.69
3.50
95.26
80.00
err_tpeak (h)
NS (%)
PC (%)
4.00
96.44
86.51
0.50
97.32
87.46
4.82
3.50
97.35
91.40
3.26
2.00
97.98
92.05
December 99
4.10
4.50
95.68
42.98
4.50
4.50
97.61
69.56
December 00
9.10
8.50
96.24
75.95
4.67
7.50
97.32
84.85
0.50
78.80
64.70
9.35
0.00
90.33
85.78
2.50
98.17
84.20
0.90
2.50
97.75
82.00
April 01
13.8
November 05
1.22
3rd
December 05
13.46
6.50
74.61
39.25
7.42
6.50
87.32
72.46
5th December 05
10.60
3.50
90.50
63.89
6.76
3.50
94.71
81.07
30th December 05 February 06 Mean absolute value
2.53
0.00
97.76
77.28
2.69
0.00
98.77
88.29
10.00
2.00
90.68
39.90
5.69
2.00
91.32
49.58
8.11
3.96
92.02
67.39
4.65
3.04
95.35
80.92
time of 3.0 h, the flood event on 3 December 2005 character-
model would be poorer in forecasting the flow depth when
ized by a complex shape of the peak region and the two flood
that event is associated with significant lateral flow. Although
events on December 1999 and February 2006 could not be
the error update model can, to some extent, improve the
successfully forecasted as reflected by their PC estimates
forecasts in the event of experiencing lateral flow, it may not
(o50%). However, for the last two events, significant lateral
give reliable forecasts when there is significant lateral flow in
flows (425% of inflow hydrograph volume) affected the model
the reach. The minimum PC estimated for the forecasted events
performance. As the proposed forecasting model has been
is greater than 60%, except for three events (December 1999,
developed using the assumption of no lateral flow in the
3 December 2005 and February 2006), out of which two events
considered reach, it is expected that the efficiency of the
are characterized by significant lateral flow.
5.0
4.5
November 1997
4.5
4.0
4.0
3.5 stage (m)
stage (m)
December 1996
3.5 3.0 2.5 2.0
inflow hydrograph observed outflow forecast outflow (1.0 hour) forecast outflow (1.5 hours) forecast outflow (2.0 hours) forecast outflow (2.5 hours) forecast outflow (3.0 hours)
1.5
3.0 2.5 inflow hydrograph observed outflow forecast outflow (1.0 hours) forecast outflow (1.5 hours) forecast outflow (2.0 hours) forecast outflow (2.5 hours) forecast outflow (3.0 hours)
2.0 1.5 1.0
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 time (hours) Figure 3 9 December 1996 event: comparison between observed and forecast stage hydrographs for different lead times at Ponte Felcino section. The input stage hydrograph at Pierantonio site is also shown.
8
10 12 14 16 18 20 22 24 26 28 30 32 34 36 time (hours)
Figure 4 9 As Figure 3, but for the event of November 1997.
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
159
3.0
Hydrology Research 9 42.2–3 9 2011
December 1999
2.8 2.6 stage (m)
stage (m)
2.4 2.2 2.0 inflow hydrograph observed outflow forecast outflow (1.0 hour) forecast outflow (1.5 hours) forecast outflow (2.0 hours) forecast outflow (2.5 hours) forecast outflow (3.0 hours)
1.8 1.6 1.4 1.2 15
20
25
30
35
40
45
50
55
7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5
November 2005 (flooding)
inflow hydrograph observed outflow forecast outflow (1.0 hour) forecast outflow (1.5 hours) forecast outflow (2.0 hours) forecast outflow (2.5 hours) forecast outflow (3.0 hours)
8
60
12
16
20
24
time (hours)
28
32 36 40 time (hours)
44
48
52
56
60
Figure 6 9 As Figure 3, but for the event of November 2005.
Figure 5 9 As Figure 3, but for the event of December 1999.
Further, Figures 7 and 8, illustrating the comparison
rising limb suddenly decreases, and at the flood peak region,
between the observed downstream stage hydrograph and
providing overestimates of the forecasted stage around this
those forecasted by both the VPMS and STAFOM models
time zone. This was observed for almost all the events studied
for the flood events that occurred on April 1997 and April
and can be seen from Figures 3–6 and from the forecast
2001, with lead times of 1.0 and 3.0 hours, reveal that the
results of other events (not shown here). Figure 9 illustrates a typical comparison between the
VPMS model has a comparable accuracy with STAFOM in
observed stage hydrograph and those forecasted by the real-
flood-stage forecasting. In order to investigate the role of the error updating
time VPMS model with and without considering the error
model in the assessment of the forecasted stage, a compara-
updating model (Equation (13)) for the floods that occurred
tive analysis was carried out between the results obtained by
on December 1996 and November 1997 with a lead-time
the proposed approach and that by Equation (12), neglecting
of 3 hours. It can be seen that the forecasting error,ef;ðjDtþTL Þ ,
the term quantifying the error of forecast, ef;ðjDtþTL Þ . The
has an important role within the forecasting procedure,
analysis showed that the adjustment due to the error updating
significantly improving the forecasting accuracy during
model is particularly significant during the advancement of
the advancement of rising limb and, also, as underlined
rising limb of the hydrograph, when the rate of increase of
above, producing an overestimation during the peak phase.
(b)
(a) 5.5
4.5
4.5
4.0
4.0
3.5
3.5
3.0 2.5 2.0
1.0
3.0 2.5 2.0
inflow hydrograph observed outflow forecast outflow (VPMS model) forecast outflow (STAFOM model)
1.5
April 1997 forecast lead-time = 3 hour
5.0
stage (m)
stage (m)
5.5
April 1997 forecast lead-time = 1 hour
5.0
inflow hydrograph observed outflow forecast outflow (VPMS model) forecast outflow (STAFOM model)
1.5 1.0
0.5
0.5 8
12
16
20
24
28
32 36 40 time (hours)
44
48
52
56
60
8
12
16
20
24
28
32 36 40 time (hours)
44
48
52
56
60
Figure 7 9 April 1997 event: stage forecasting by the VPMS and the STAFOM models for two lead times of a) 1 hour and b) 3 hours at Ponte Felcino section. The input stage hydrograph at Pierantonio site is also shown.
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
160
Hydrology Research 9 42.2–3 9 2011
(a)
(b) 4.0
April 2001 forecast lead-time = 1 hour
3.5
3.5
3.0
3.0 stage (m)
stage (m)
4.0
2.5
April 2001 forecast lead-time = 3 hour
2.5 2.0
2.0 inflow hydrograph observed outflow forecast outflow (VPMS model) forecast outflow (STAFOM model)
1.5
inflow hydrograph observed outflow forecast outflow (VPMS model) forecast outflow (STAFOM model)
1.5 1.0
1.0 5
10
15
20
25
30
35
40
45
50
5
time (hours)
10
15
20
25
30
35
40
45
50
time (hours)
Figure 8 9 As Figure 7, but for the event of April 2001.
The significant effect of the updating error technique may be
model has the potential for practical forecasting applications
attributed to the consideration of the simplified model struc-
in hydrometric data-based modelling provided that the
ture and the assumption introduced.
adopted forecasting lead time is not longer than the mean wave travel time of the selected river reach, which for the investigated case study can be assumed equal to 1.5–2.0
CONCLUSIONS
hours. Further investigations on different case studies have to be carried out in order to verify the proposed forecasting
The application of a VPMS hydrograph routing method for
model accuracy and, furthermore, it would be advisable to
real-time flood forecasting at a river gauging site is demon-
extend the model formulation to take into account significant
strated in this study. Based on the forecasting performance for
lateral flow contribution entering along the selected river
several investigated events, one can infer that the proposed
reach.
(a)
(b) 5.0
5.0 December 1996 forecast lead-time = 3 hour
4.0
4.0
3.5
3.5
3.0
3.0
2.5 2.0
2.5 2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
a)
-1.0
November 1997 forecast lead-time = 3 hour
4.5
stage (m)
stage (m)
4.5
b)
-1.0 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 time (hours)
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 time (hours)
Figure 9 9 Comparison between the observed stage hydrograph and those forecast by the VPMS model with and without the error updating technique for a lead-time of 3 hours for the flood events that occurred on a) December 1996 and b) November 1997 at Ponte Felcino section. The error of forecast computed by Equation (13) is also shown.
161
M. Perumal et al. 9 Real-time flood stage forecasting by VPMS method
ACKNOWLEDGEMENTS The authors are grateful for the financial support from the ‘‘International short-term mobility programme for scientists/ researchers from Italian and foreign institutions’’ granted to the first author by the CNR (National Research Council) of the Italian Government to carry out part of this research at the CNR–IRPI Office of Perugia, Italy. The authors are also grateful to Umbria Region for providing most of the analysed data.
REFERENCES Ahsan, M. & O’Connor, K. M. 1994 A reappraisal of the Kalman filtering technique, as applied in river flow forecasting. J. Hydrol. 161, 197–226. Apollov, B. A., Kalinin, G. P. & Komarov, V. D. 1964 Hydrological Forecasting (translated from Russian). Israel Programme for Scientific Translations, Jerusalem. ASCE Task Committee on Definition of Criteria for Evaluation of Watershed Models of the Watershed Management Committee, Irrigation and Drainage Division 1993 Criteria for evaluation of watershed models. J. Irrig. Drain. Eng. ASCE 119(3), 429–442. Babovic, V., Canizares, R., Jensen, H. R. & Klinting, A. 2001 Neural networks as routine for error updating of numerical models. J. Hydraul. Eng. ASCE 127(3), 181–193. Barbetta, S., Brocca, L., Melone, F. & Moramarco, T. 2008 On the lateral inflows assessment within a real-time stage monitoring addressed to flood forecasting. In: Proceedings of the iEMSs Fourth Biennial Meeting: International Congress on Environmental Modelling and Software (iEMSs 2008) (Sa`nchez-Marre`, M., Be´jar, J., Comas, J., Rizzoli, A. E. & Guariso, G. (eds.)). International Environmental Modelling and Software Society, Barcelona, pp. 438–445 Box, G. E. P. & Jenkins, G. M. 1970 Time Series Analysis, Forecasting and Control. Holden-Bay Inc, San Francisco. Cunge, J. A. 1969 On the subject of a flood propagation method (Muskingum Method). J. Hydraul. Res., IAHR 7(2), 205–230.
Hydrology Research 9 42.2–3 9 2011
Dooge, J. C. I., Strupczewski, W. G. & Napiorkowski, J. J. 1982 Hydrodynamic derivation of storage parameters of the Muskingum model. J. Hydrol. 54(4), 371–387. Gelb, A. (ed.) 1974 Applied Optimal Estimation. MIT Press, Cambridge, MA. Huang, W.-C. 1999 Kalman filter effective to hydrologic routing? J. Marine Sci. Tech. 7(1), 65–71. Moramarco, T., Barbetta, S., Melone, F. & Singh, V. P. 2006 A real time stage Muskingum forecasting model for a site without rating curve. Hydrol. Sci. J. IAHS 51(1), 66–82. Nash, J. E. & Sutcliffe, J. V. 1970 River flow forecasting through conceptual models, Part-1: A discussion of principles. J. Hydrol. 10(3), 282–290. Neal, J. C., Atkinson, P. M. & Hutton, C. W. 2007 Flood inundation model updating using an ensemble Kalman filter and spatially distributed measurements. J. Hydrol. 336(3–4), 401–415. Perumal, M. 1994a Hydrodynamic derivation of a variable parameter Muskingum method: 1. Theory and solution procedure. Hydrol. Sci. J. IAHS 39(5), 431–442. Perumal, M. 1994b Hydrodynamic derivation of a variable parameter Muskingum method: 2. Verification. Hydrol. Sci. J. IAHS 39(5), 443–458. Perumal, M. & Ranga Raju, K. G. 1998a Variable-parameter stage– hydrograph routing method. I: Theory. J. Hydrol. Engg. ASCE 3(2), 109–114. Perumal, M. & Ranga Raju, K. G. 1998b Variable-parameter stage– hydrograph routing method. II: Evaluation. J. Hydrol. Engg. ASCE 3(2), 115–121. Perumal, M. & Ranga Raju, K. G. 1999 Approximate convection diffusion equations. J. Hydrol. Engg. ASCE 4(2), 161–164. Perumal, M., Moramarco, T., Sahoo, B. & Barbetta, S. 2007 A methodology for discharge estimation and rating curve development at ungauged river sites. Water Resour. Res. 43(2), W02412, 1–22. Perumal, M., Moramarco, T., Sahoo, B. & Barbetta, S. 2010 On the practical applicability of the VPMS routing method for rating curve development at ungauged river sites. Water Resour. Res. 46(3), W03522. Refsgaard, J. C. 1997 Validation and intercomparison of different updating procedures for real-time forecasting. Nordic Hydrology 28, 65–84.
First received 29 May 2009; accepted in revised form 6 December 2009. Available online February 2011
